Math 6510 • Differentiable Manifolds

Let \(M,N\) be two closed connected oriented \(n\)-manifolds and
\(f:M\to N\) a degree 1 map. (Recall that this means that \(f_*[M]=[N]\) where
\([M],[N]\) are the fundamental classes of \(M,N\) respectively.)
\begin{enumerate}
\item[(a)] Show that \(f_*:H_k(M)\to H_k(N)\) is onto for every \(k\). Hint:
\(f_*(\alpha\cap f^*(z))=f_*(\alpha)\cap z\). Explain.
\item[(b)] Deduce that if \(M=S_g\), \(N=S_h\) are surfaces of genus \(g\), \(h\)
respectively, then \(g\ge h\).
\end{enumerate}